On Walsh series with monotone coefficients

We prove that if $a_n\downarrow 0$ and $\sum_{n=0}^\infty a_n^2=+\infty$ then the Walsh series $\sum_{n=0}^\infty a_nW_n(x)$ has the following property. For any measurable function $f(x)$ which is finite almost everywhere, there are numbers $\delta_n=0,\pm 1$ such that the series $\sum_{n=0}^\infty\delta_na_nW_n(x)$ converges to $f(x)$ almost everywhere. This assertion complements and strengthens previously known results about universal Walsh series and Walsh null-series.